Rotations and Reflections

Time Limit: 2000/1000 MS (Java/Others)

Memory Limit: 65536/32768 K (Java/Others)


Many games, tricks and puzzles depend on determining whether two patterns on a rectangular grid are the ``same'' or not. For instance, the 96 different ways of arranging 8 queens safely on a chessboard can be shown to consist of rotations and/or reflections of only 12 basic patterns.

Write a program that will read in pairs of patterns and determine whether there is a simple transformation that will convert one into the other. Because symmetrical patterns bear many relationships to each other, the transformations must be checked in a specific order. The possible transformations (in order) are:

Preservation:           The patterns are identical

90 degree rotation :    The pattern was rotated clockwise by 90 degrees

180 degree rotation:    The pattern was rotated clockwise by 180 degrees

270 degree rotation:    The pattern was rotated clockwise by 270 degrees

Reflection:             The pattern was reflected about the horizontal axis (effectively by a mirror held at the top of the pattern)

Combination:            A reflection (as above), followed by one of the above rotations

Improper:               The patterns do not match under any of the above transformations


Input will consist of a series of pairs of patterns. Each set will consist of a line containing a single integer N (2 <= N <= 10) giving the size of the patterns, followed by N lines. Each line will consist of N dots or `x's (specifying a line of the original pattern), a space, and another set of N dots and `x's (specifying a line of the transformed pattern). The file will be terminated by a line consisting of a single zero (0).


Output will consist of a series of lines, one for each pattern pair in the input. Each line will consist of one of the following: `Preserved', `Rotated through m degrees' (where m is one of 90, 180 or 270), `Reflected', `Reflected and rotated through m degrees', `Improper'.

Sample Input

5 x...x ....x .x... ...x. ...x. .x... ..x.x ..x.. ....x xx..x 2 x. xx x. xx 4 ..x. ...x xx.. .... .... xx.. ...x ..x. 4 .x.. ..x. .x.x x... .... ..xx ..x. .... 0

Sample Output

Rotated through 90 degrees Improper Reflected Reflected and rotated through 270 degrees